Implicit Integration of the Unified Yield Criterion in the Principal Stress Space

An implicit numerical integration algorithm is presented for the unified yield criterion, which could encompass most other yield criteria. The modification matrix, which is used to convert the continuum tangent modular matrix into the consistent tangent modular matrix, is derived for the return to planes, lines, and the apex of the unified yield criterion with multisurface plasticity with discontinuities. The stress update and consistent tangent modular matrix are first derived in closed form in the principal stress space, and then they are transformed back into the general stress space by coordinate transformation. Three numerical examples are used to demonstrate the effectiveness of the presented algorithm. The correctness of the developed algorithm is validated by the analytical solution and ABAQUS solution with the built-in Mohr-Coulomb model. The developed algorithm is also demonstrated to be least twice more efficient than the ABAQUS built-in algorithm. The presented algorithm for the unified yield criterion can facilitate the understanding of the effect the intermediate principal stress. With the increase in b, the force versus deflection curve at the midspan increases for the beam under three-point bending, and the critical radius of the elastoplastic interface decreases (i.e., the plastic zone becomes small) for the circular tunnel under hydrostatic pressure.